Testbook group theory question

Question

Let G be a group of order 49. Then

• G is abelian

• G is cyclic

• G is non-abelian

• Centre of G has order 7

Comments

i didn’t get the formula

Answer 1

Theorem :-  Group with $P^2$ element ,where p is a prime number is always abelian group .

Now here order of group ,$O(G)=49=7^{2}$ 

Now $7$ is a prime number so the group will be abelian .

this theorem is one way implication :-

The order of group is $P^2$ $\Rightarrow$ the group is abelian .

The group is abelian does not implies the order of group is $P^2$ .

This group of order 49 may or may not be cyclic as if a goup is abelian does not mean the group should be cyclic as well .

 

Reference :- https://math.stackexchange.com/questions/64371/showing-group-with-p2-elements-is-abelian?noredirect=1&lq=1

 

Answer 2

There was given a theorem :

A group G is abelian when O(G) is in the form of O(G)=p1^m1 p2^m2 … pn^mn

 where, p1, p2, …. Pn are the prime numbers

Comments

Sir could you please give any example for the theorem which you have mentioned.